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48                        ANALYTICAL MECHANICS
The resultant of a system of forces consists, therefore*, of a single force and a single torque which, when reversed, will keep the rigid body in equilibrium against the action of the given system of forces.
55. Resultant of Coplanar Forces Acting upon a Rigid Body. -Let FI, F2, . . . Fn denote the given forces and let the .r//~ plane be their plane of action. Then, if R, X, and Y denote the resultant force and its components, respectively, we have
and             tan0 = —>                                                   (IX)
where the terms in the right-hand members of the first two equations are the components of the given forces, and 0 is the angle R makes with the #-axis.
On the other hand if G0 denotes the resultant torque and di, dz, . . . , dn denote the distances of the origin from the lines of action of the forces, then
00 = Fi4 + ft4 + • - • +/<Vi«                  (X)
If we represent this torque by the moment of the resultant force about the #-axis, then

gives the distance of the line of action of the resultant force from the origin.
Find the resultant of the six forces acting along the sides of the hexagon of Pig. 36.
Taking the sum of the components along the x and y dircctioiw, wo have